| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Area transformation under matrices |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring (a) solving two simultaneous equations from matrix multiplication, and (b) applying the determinant-area scaling rule. Both parts are standard textbook exercises with no novel insight required, though being Further Maths content places it slightly above average A-level difficulty. |
| Spec | 4.03i Determinant: area scale factor and orientation4.03r Solve simultaneous equations: using inverse matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix}-4 & a \\ b & -2\end{pmatrix}\begin{pmatrix}4 \\ 6\end{pmatrix} = \begin{pmatrix}2 \\ -8\end{pmatrix}\) | M1 | Using information to form the matrix equation. Can be implied by both correct equations below |
| \(-16 + 6a = 2\) and \(4b - 12 = -8\) | M1 | Any one correct equation; any correct horizontal line |
| \(a = 3\) and \(b = 1\) | A1 | Any one of \(a=3\) or \(b=1\) |
| A1 | Both \(a=3\) and \(b=1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\det \mathbf{A} = 8 - (3)(1) = 5\) | M1 | Finds determinant by applying \(8 -\) their \(ab\) |
| A1 | \(\det \mathbf{A} = 5\) | |
| Area \(S = 5 \times 30 = 150\) (units)\(^2\) | M1 | \(\frac{30}{\text{their } \det\mathbf{A}}\) or \(30 \times (\text{their } \det\mathbf{A})\) |
| A1\(\checkmark\) | 150 or follow-through answer. If \(\det A < 0\) allow ft provided final answer \(> 0\) |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}-4 & a \\ b & -2\end{pmatrix}\begin{pmatrix}4 \\ 6\end{pmatrix} = \begin{pmatrix}2 \\ -8\end{pmatrix}$ | M1 | Using information to form the matrix equation. Can be implied by both correct equations below |
| $-16 + 6a = 2$ and $4b - 12 = -8$ | M1 | Any one correct equation; any correct horizontal line |
| $a = 3$ and $b = 1$ | A1 | Any one of $a=3$ or $b=1$ |
| | A1 | Both $a=3$ and $b=1$ |
**(4 marks)**
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\det \mathbf{A} = 8 - (3)(1) = 5$ | M1 | Finds determinant by applying $8 -$ their $ab$ |
| | A1 | $\det \mathbf{A} = 5$ |
| Area $S = 5 \times 30 = 150$ (units)$^2$ | M1 | $\frac{30}{\text{their } \det\mathbf{A}}$ or $30 \times (\text{their } \det\mathbf{A})$ |
| | A1$\checkmark$ | 150 or follow-through answer. If $\det A < 0$ allow ft provided final answer $> 0$ |
**(4 marks)**
---5. $\mathbf { A } = \left( \begin{array} { r r } - 4 & a \\ b & - 2 \end{array} \right)$, where $a$ and $b$ are constants.
Given that the matrix $\mathbf { A }$ maps the point with coordinates $( 4,6 )$ onto the point with coordinates $( 2 , - 8 )$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$.
A quadrilateral $R$ has area 30 square units.\\
It is transformed into another quadrilateral $S$ by the matrix $\mathbf { A }$.\\
Using your values of $a$ and $b$,
\item find the area of quadrilateral $S$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2011 Q5 [8]}}